BCom-Business Statistics 239

8/16/2019 BCom-Business Statistics 239

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B.Com.

Second Year

Paper No.VIII

BUSINESS STATISTICS

BHARATHIAR UNIVERSITY

SCHOOL OF DISTANCE EDUCATION

COIMBATORE – 641 046


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BCom-Busines Statistic

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CONTENT

Lessons PAGENo.

UNIT-I

Lesson 1 Statistics – Meaning and Scope 7

Lesson 2 Characteristics and Limitations 15

Lesson 3 Presentation of Data(Diagrams and Graphs)

20

Lesson 4 Frequency Distributions and Charts 35

Lesson 5 Measures of Central Tendency 51

UNIT-II

Lesson 6 Measures of Dispersion(The Range and the Mean Deviation)

75

Lesson 7 Measures of Dispersion(The Standard Deviation and the Quartile Deviation)

85

Lesson 8 Measures of Skewness 103

UNIT-III Lesson 9 Regression Analysis 111

Lesson 10 Correlation Analysis 126

UNIT-IV

Lesson 11 Construction of Index Numbers 149

Lesson 12 Construction of Index Numbers(Price and Quantity Relatives)

156

Lesson 13 Construction of Index Numbers

(Composite Index Numbers)

161

Lesson 14 Consumer Price Index Numbers 180

UNIT-V

Lesson 15 Time Series Analysis 195

Lesson 16 Seasonal Variations and Forecasting 215

Lesson 17 Sampling Methods 231


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(Syllabus)

PAPER – VIII BUSINESS STATISTICS

Objectives : To promote the skill of applying statistical techniques in business.

UNIT-I

Meaning and Scope of Statistics – Characteristics and Limitations – Presentationof Data by Diagrammatic and Graphical Methods – Measures of Central

Tendency – Mean, Median, Mode Geometric Mean, Harmonic Men.

UNIT-II

Measure of Dispersion and Skewness – Range, Quartile Deviation and StandardDeviation – Pearson’s and Bowley’s Measures of Skewness.

UNIT-III

Simple Correlation – Pearson’s coefficient of Correlation – Interpretation of Co-efficient of Correlation – Concept of Regression Analysis – Coefficient ofConcurrent Deviation.

UNIT-IV

Index Numbers (Price Index Only) – Method of Construction – Wholesale andCost of Living Indices, Weighted Index Numbers – LASPEYRES’ Method,

PAASCHE’S Method, FISHER’S Ideal Index. (Excluding Test of Adequacy of IndexNumber Formulae)

UNIT-V

Analysis of Time Series and Business Forecasting – Methods of Measuring Trendand Seasonal Changes (Including Problems) Methods of Sampling – Samplingand Non-Sampling Errors (Theoretical Aspects Only)

Note – Distribution of Mark : Theory : 20 % Problems – 80 %

Book for Reference

1. Navanitham, P.A., “Business Mathematics and Statistics”, Jai Publishers, Trichy, 2004.

2. S.P.Gupta, “Statistical Methods”,

3. M.Sivathanu Pillai, “Economic and Business Statistics”.


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UNIT – I


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LESSON-1

STATISTICS – MEANING AND SCOPE

CONTENTS

1.0. Aims and Objectives

1.1. Meaning of Statistics

1.2. Statistical Investigation

1.3. Scope of Statistics

1.4. Summary

1.5. Lesson End Activity

1.6. Points for Discussion

1.7. Suggested Reading/Reference/Sources

1.0 AIMS AND OBJECTIVES

This lesson aims to provide in general the meaning and definition of statistics, and their

role in various disciplines and different phases of human endeavour. The significance ofstatistical theory is highlighted. The need of statistical investigation in making vital

decisions about the universe or the population under study is also presented.

1.1 MEANING OF STATISTICS

Statistics is a term which has several meanings in practice. The word ‘statistics’ can beused in two senses, namely, (a) to describe values which summarize data, such as

percentages or averages and (b) to describe the topic of statistical method.

The term ‘data’ would mean facts or things certainly known from which conclusions may be drawn. Statistics is regarded commonly as data which is defined as a collection of

information on certain variables or characteristics such as the prices of commodities

during a particular period, the number of business enterprises in a city, the number offinancial institutions in a state, illiteracy level of population in a district, health conditions

of people, geographical locations, weather conditions during a period of time etc.

Statistical method can be described as (a) the selection, classification and organisation of basic facts into meaningful data, and as (b) summarizing, presenting and analysing the

data into useful information.


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Statistics, in general, is defined in many ways; few of them are presented below:

It is the aggregate of facts and figures.

• It stands for record of numerical facts and figures.

• It is termed as statistical methods that are described for the principles and techniquesapplied in the collection, analysis and interpretation of data on the statements offacts.

• It is a field concerned with scientific methods for collecting, organizing,summarizing, presenting and analyzing data, as well as drawing valid conclusionsand making reasonable decisions on the basis of such analysis.

• It is a body of concepts and methods used to collect and interpret data concerning a particular area of investigation and to draw conclusions in situations whereuncertainty and variations are present.

• It is a field which refers to the science and art of obtaining and analyzingquantitative data with a view to make sound inference in the face of uncertainty.

1.2 STATISTICAL INVESTIGATION

The statistical information obtained from many different sources is being used by business

establishments to make vital decisions about various business and managerial problems based on one or more techniques, described as statistical methods. The decisions thatwould be taken are the outcomes of an activity, called statistical investigation. In general,

a statistical investigation is defined as a process or a set of processes of studying the

population under study with reference to one or more characteristics using statistical data.

A few examples of statistical investigation are listed below:

1. Assessing people’s opinion on the choice of various schemes proposed by business

firms to market their products.

2. Assessing people’s preference on the choice of candidates contesting in elections.

3. Studying the impact of economic policies adopted by the government on labourforce.

4. Forecasting the production of items during a particular future period.

5. Studying the mental depression and stress of managers of business firms.

Statistical investigations are usually undertaken to make decisions on population with

reference to one or more characteristics of interest which may be inadequate or

unobtainable. In situations involving costly or destructive nature of items or timeconsuming activities or problems, investigations will be based on sample information that

would be drawn by specific procedures from the population.


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The elements of any statistical investigation or study are classified into four, namely,

1. Specification of objectives, (2) gathering of information, (3) analysis of data and (4)

statements of findings, which are described briefly as follows:

A statistical study is generally carried out by specifying the objectives of the study. With

reference to the specified object and its scope, the relevant information necessary for thefulfilment of the purpose will be found out. An object which is not explained precisely

will create difficulty and confusion and with that only data which may not be relevant to

the purpose will be resulted. Great care should be attached in defining all the aspects of

the problem so that the stated objective will be met.

The second element is the collection of information or data relevant to the objective of the

study. This may be done by direct observation or by conducting experiments or byreferring to official, historical, authentic records or by conducting surveys. Generally,

information takes the form of numerical measurements of certain characteristics or the

record of the possession of attributes, such as sex of the people, habit of the people etc.

The third element is the analysis of data, which is considered as the process of applyingappropriate statistical methods to the data collected for the specific objective and

extracting information relevant to the problem under study.

The fourth element is making statements of findings on the problems raised in thespecification of objectives. As per the findings it may be possible to retain the existing

theory or to suggest a new theory to explain certain situations.

1.3 SCOPE OF STATISTICS

Statistics is playing an increasingly important role nearly in all phases of human

endeavour. It deals not only with affairs of the state, but also with many other fields suchas agriculture, biology, business, chemistry, commerce, communications, economics,

education, electronics, medicine, physics, political science, psychology, sociology and

numerous other fields of science and engineering.

A detailed discussion of the need and scope of statistics in other branches of science,humanities and social sciences and engineering is presented below:

Statistics Simplifies Complex Problems

Statistics is much important in every sphere as it simplifies complexity. The facts and

figures which constitute statistical data can not be assimilated just by looking at them. Thestatistical methods effectively make these data as simple as possible so that they are

intelligible easily and readily understandable, which would provide a great service to find

solutions to complex problems. Statistical methods describe a phenomenon in a verysimple way. For instance, suppose that one is interested to study the economic system of a

country. The system can not be understood simply by a descriptive way, which does not

use statistical information. It is known that any physical and random phenomena can beexpressed quantitatively. Thus, whenever it is possible to express the various aspects of


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the economic system as numeric measures, the system could be understood without any

difficulty and ambiguity.

Statistics Measures and Highlights Results

Statistical methods provide the ways and means of measuring the results of various

policies on economics, trading, banking etc. For instance, the effect of a rise in the bankrate against loan to be given to the industries can be studied in a proper manner by means

of a statistical study of the phenomenon. Though it is a complex exercise, the statistical

methods help to render service to a great extent to ease out the difficulty. The statisticalideas will further help to measure whether a rise in the bank rate has affected the

industries adversely or favourably by taking into consideration a comparative study of the present situation with the past. The statistical thinking further helps to make a decision

whether the change has been beneficial or otherwise from the point of view of industries.All such measures and decisions could be made possible only with the use of adequate

statistical data.

Statistics Studies Relationships Among Phenomena

Statistical methods render a service in studying the relationship that exists between two or

more phenomena. In all types of economic and business studies the importance ofobserving relationship between different phenomena is very great. For instance, the

relationship between, say, price and supply or demand and price of a commodity is a

phenomenon which requires a very careful and close study before any generalization can be made. In the absence of statistical methods it would be very difficult to arrive at a

precise and correct conclusion in this respect.

Statistics Deals with Human Experience

The experience and knowledge gained by human can be enhanced and assimilated by thescience of statistics so as to easily understand, describe and measure the effects of the

actions taken by him or by others. The science has provided vital methods, which can beused anywhere and study any problems which deal with deterministic and random

phenomena in correct perspectives and on the right directions.

The following discussions indicate how statistics is indispensable in different branches of

human activities:

Statistics and their Relationships with the Common Man

The science of statistics is important to common man in every walk of his life. It has theuniversal applicability in all the fields where the human steps in. Millions of people all

over the world use statistics in their day-to-day actions though they might not have heard

the term ‘statistics’. While making decisions on various problems in different situations, ahuman makes use of information which he gets from the universe or population. For

instance, suppose that a person wishes to invest his earnings in stocks. Before taking a

decision on the choice of companies, number of shares to purchase, amount of investmentetc., he gets detailed statistical information such as the market fluctuations of shares, the

performance of the company in the past. A thorough analysis of data in such cases will


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help the person to make effective decisions. As another example, consider a farmer who

wishes to have a particular quantity of rain in a particular season so that he may have agood crop. Here, based on his past experience in crop cultivation and seasonal changes he

would have an idea of the correlation that exists between rainfall and crop yields.

Importance of Statistics in Theory of Economics

Economics and statistics are in fact inseparable. Most of the concepts in economics can be

treated using statistical relationships through statistical models. Almost all economics

problems are studied and compared with the help of statistical data. The purchasing powerof people, consumption behaviour, income and expenditure on certain goods are analyzed

using statistical data. Economic policies, reforms and their impacts on the society are being studied based on statistical information. Statistics of production, exchange and

distribution describe the wealth of the nation, development of the nation and distributionof national dividend. All such statistics are needed to study about the progress and growth

of the economy of the country. Thus, in all types of economic problems statisticalapproach is essential and statistical analysis is much useful. Mathematics, statistics and

accounting are the powerful instruments which help the modern economist to increase and

improve economic growth.

Statistics and their Significance in Planning

For the development of any country or state, planning is essential. The schemes of the

government are based on planning. Planning cannot be imagined without statistics. Forinstance, growing population and growing demand of commodities are a major concern

for many under developed and developing countries. In order to control population and tomeet the demand, a state or a government needs proper planning, which obviously use

statistical information. In order that any planning is to be successful, statistical data, morecomplex in nature, should be analyzed carefully and correctly. Various countries

implement the economic plans only by conducting statistical studies of the economic

resources of the respective countries and by finding the possible ways and means ofutilizing these resources in the best possible manner. Various plans that have been

prepared for the economic development of India have also made use of the statistical

material available about various economic problems.

Statistics and Commerce

Statistics is an important aid to business and commerce. In any business establishment,forecasts are made based on the past performance of the firm. Success or failure in

business is realized according as the forecasts made prove to be accurate or otherwise. A

business man, who uses the forecasting tool to plan for the future, succeeds in businesswhen the result of forecasting is precise and accurate. A business man fails in his business

due to wrong expectations and calculations, which arise due to faulty reasoning and

inaccurate analysis of various causes affecting a particular phenomenon. Modern devices,

called economic barometers, considered to be the statistical methods, being applied by the business people have made business forecasting more definite and precise. Analysis of

demand of goods, supply of commodities, the prices, effect of trade cycles and seasonal


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fluctuations help a businessman to take final decision about the productivity and demand.

All these aspects are carried out using the statistical principles. The effects of booms anddepressions are to be considered seriously by a businessman to succeed in business. Such

effects are being analysed only by statistical concepts using information. A study of all

these things is in reality a study of statistics and hence we say that all types of businessmen have to make use of statistics in one form or the other if they want any

success in their profession.

Statistical data are used extensively by promoters of new business so as to arrive atdecisions about starting a new firm.

The methods of statistical analysis are particularly appropriate in finding the solution of problems connected with the internal organization and administration of business units

and with the processes of buying and selling that bring the businessman into contact withthe price system. Various branches of commerce, such as cost accounting utilise the

services of statistics in different forms. For instance, the technique with the help ofstatistical methods helps producers to decide about the prices of various commodities.

Similarly, promoters of new business make extensive use of statistical data to arrive at

conclusions which are vital from the point of view while starting a new concern.

Application of Statistics in Business Management

Managers in business firms always need to make decisions in the face of uncertainty. The

statistical tools such as collection, classification, tabulation, analysis and interpretation of

data deal with the problem of uncertainty and are found to be useful in making wise

decisions at various levels of managerial function.The production programming, quality and inventory control are the statistical tools which

are applied to the problems concerned with business management. The production

programming techniques depend on quality of sales forecasts and projections. The sales

forecasts are made using statistical data, which provide sales estimates. Effective controlon sales is done based on a statistical study of trend. Market research, consumer preference studies, trade channel studies and readership surveys are other methods of sales

control which make an extensive use of statistical tools.

Statistical methods also come to the aid of quality control. Here, random sampling method

is adopted to decide whether a lot of items supplied by a manufacture is of standard

quality or not.Inventory control is essential for economical functioning of business enterprises. It relates

both to quantitative and qualitative aspects. The stocking of inventories at the optimum

level depends on the accuracy of sales forecasts and correlation between the final productand size and quantity of each raw material, tools, equipment, fuel, etc., needed for it.

Quality Control on inventory is not only facilitated but also made more accurate with the

aid of statistics. Here again the method of random sampling is adopted in choosing the

items from a lot of items for inspection. The whole lot is accepted if the sampled items areconforming to specifications. The procedure may be a complicated one when it is required

to inspect each and every item of inventory purchases.


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Significance of Statistics to the States or Countries

Economic planning and development for the welfare of the people of a state are usuallydone with statistical data. States use extensively the data in their administration. States

propose new schemes for the people. Most often they need to examine or foresee the kind

of impact of the scheme on the people if the schemes are implemented. This exercise can

be done only with the help of numerical data. Statistical investigation is being carried out by the governments to find the solution or remedies to the social problems which erupt in

the states. The states often get data from their departments and various other sources and

use them for various purposes. For instance, based on the data it collects, a state can havean idea of the literacy level, the need of the facility, the requirements of funds for various

department proposals etc. For every scheme to be implemented in the states, the

governments want to have estimates of fund requirements. This is done using statistical

facts and figures. In the economic area, for finding out the prosperity of the country thecentral government wishes to estimate the figures of national income. Though a state is an

administrative body, it carries on businesses of various kinds and has monopoly in manycases. For instance, public transport system and co-operative stores are being supported by

the governments. In order to carry on business houses which the state holds in its control,

it needs statistics.

Application of Statistical Methods in Research

Most of the modern statistical methods and statistical information play a vital role inresearch in different fields of science, engineering, medicine and social sciences. In the

field of agriculture, experimental designs are proposed and analyzed using statistical

methods to study about crop yields with different types of fertilizers and different types ofdiets and environments. In the field of medicine and public health, the statistical methods

such as clinical trials and survival analysis are used for testing the efficacy of new

medicines and methods of treatment. In the field of industry, the concepts of qualitycontrol and design of industrial experiments are applied as part of research and

development activity, which helps in improving quality and productivity. In the fields of

economics and commerce, financial data are being processed through statistical methods,which help to suggest new economic theories. Market researches are carried on by making

extensive use of statistical methods. Irrespective of any field, any researcher will always

present his findings with statistical evidence and significance as the results are mostly based on statistical information and numerical facts and figures.

Acceptability of Statistical Methods

Statistical methods have the prestige of its universal acceptability. All governments in the

world countries need statistical data for planning and implementing various schemes for

the welfare of the people. Statistical concepts assist in planning the initial observations, inorganizing them and formulating hypotheses from them, and in judging whether the new

observations agree sufficiently well with the predictions from the hypotheses. Statistical

knowledge and information of both deterministic and random nature are being used byscientists of all disciplines to propose and develop new theories. Persons from all walks of

life, astrologers, astronomers, biologists, meteorologists, botanists, and zoologists make

use of statistics and statistical methods extensively in their research. Statistics, when used


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properly and effectively, would result in a reasonable standard of accuracy of results for

the problems of nondeterministic nature. Thus, the importance, utility and indispensabilityof statistics as a branch of mathematical science to the modern world have been indicated

by its universal applicability.

1.4 SUMMARY

Statistics is concerned with data pertaining to population and deals with methods withwhich certain studies related to population are done. In this lesson, the meaning and

definition of statistics are presented. The notion of statistical investigation, which is aframework for making a study about the population based on statistical data, and its need

are described with illustrations. The importance of statistics as data and as a set of tools inhuman activities and in various other disciplines is elaborated in a separate section.

1.5 LESSON END ACTIVITY

1. Get information about the weekly sales (in Rs.) of commodities in a departmental

store near your home during the first six months in the year 2008.

2. Collect data relating to monthly income of families living in your street and theirweekly expenditure.

1.6 POINTS FOR DISCUSSION

1. Define the term ‘statistics’.

2. Explain the meaning of statistics.

3. What is meant by statistical investigation? Give illustrations.

4. Describe the importance of statistics in commerce.

5. Explain the scope of statistics in business management.

6. Discuss the need of statistics in economics and in research.

7. Explain the significance of statistics in studying problems related to various branches of sciences and humanities.

8. Elaborate the meaning and scope of statistics.

9. State the purposes which statistics serve.

1.7 SUGGESTED READING/REFERENCE/SOURCES

1. Pal, N., and S. Sarkar (2005), Statistics – Concepts and Applications, Prentice – Hall,

Englewood Cliffs, NJ, US.


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LESSON-2

CHARACTERISTICS AND LIMITATIONS

CONTENTS


2.1. Characteristics of Statistics

2.2. Limitations of Statistics

2.3. Summary





The material presented in this lesson enables one to understand the intended purpose ofstatistics and related features they should possess. By learning the contents given in this

lesson, one will be able to give a proper attention to the limitations of statistics while

applying the theoretical concepts of statistics.

2.1 CHARACTERISTICS OF STATISTICS

Statistics, in general, must possess the following chief characteristics.

1. Statistics must be numerical statements of facts

The qualitative characteristics of a population under study do not form part of statisticalstudies and hence should be expressed or reduced in terms of numerical quantities. The

characteristics such as good, average, poor are the qualitative expressions, which may be

expressed as numbers like 2, 1 and 0 respectively. For example, a good student in a classmay be assigned with the number 2, where as a poor student with 0. Similarly, the

standard of a student may be specified according to the marks he secures in a test. For

instance, when a student secures 60 per cent marks and above, he may be classified asgood.

The annual productions of cereals per acre in the previous period and in the current period

respectively reported as 40 and 55 quintals constitute statistical statements. Similarly, agesof persons A and B are specified as 20 years and 60 years make statistical statements.


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2. Statistics are aggregate of facts

Statistics do not take into account individual cases. For instance, an individual working ina firm whose average monthly income is Rs. 20,000 does not constitute statistics unless

the income details of the total number of individuals is given out. Similarly, a single ageof 25 years or 40 years is not statistics but a series relating to the ages of a group of

persons would be called statistics. Likewise, aggregates of figures relating to birth, death,

purchase, sale, etc., would be called statistics because they can be studied in relation to

each other and are capable of comparison, where as the single figure relating to birth,death, purchase, sale, etc., does not form statistics. Studies pertaining to individuals are

not significant from statistical point of view, for conclusions cannot be drawn by means of

comparison and also the figure cannot be treated otherwise. In order to advance the studyit is necessary that other observations must be made available.

3. Statistics should be capable of being related to each other

In order to understand clearly the percentage of students who have passed in anexamination it is important to know that how many students has appeared the examination

and to make comparisons it is also required to know about the figures of other sections of

the class. For example, suppose that the number of students in a class and the number ofmenial staff in the school are specified. These figures are all numerical statements of facts.

Even then, they cannot be called as statistics as there is no apparent relationship among

them,

4. Statistics must have certain object behind themThey must be collected for a pre-determined purpose. Only figures that are relevant and

relate to the objective of enquiry should be provided. Sets of figures without any object

behind them are not capable of being placed in relation to each other. Suppose that in astudy related to finding the teacher-student ratio, it is required to have information about

the number of students and number of teachers in a school. Obviously, these figures may

constitute statistics, as they are presented with an objective. It is also much important thatthe aggregates of facts must pertain to the objective of enquiry in order that they may be

designated as statistics.

5. Statistics are affected to a marked extent by a large number of causes

Usually, statistical facts are not traceable to a single cause. It is known that the demand ofa commodity depends on the supply of the commodity. As the supply of the commodity

decreases, the demand increases. But, in reality the change in the demand is not only

caused by the supply, but also other factors such as the price of the commodity, people’s

choice, prices of related commodities etc. Similarly, statistics of prices are affected byconditions of supply, demand, exports, imports, currency circulation and a large number

of other factors. Thus, there are many factors which influence changes in a variable under

study and there should not be only a single factor responsible for bringing about a changein the variable. When there is only one factor operating at a time, the study ceases to be

significant from statistical point of view.


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6. Statistics should exhibit a reasonable standard of accuracy

While collecting statistical information one should be cautious so as to get or maintain areasonable standard of accuracy. As statistics, sometimes, deal with large numbers, it

becomes impossible to observe each one of the items individually. Therefore, it becomes

necessary to observe and analyze a sample of items and to apply the result to the entiregroup, called population. Usually, in such cases population characteristics can only be

estimated from sample information. Obviously, the estimated figures cannot be absolutely

accurate and precise and the degree of accuracy expected in such figures depends to a

large extent on the purpose for which statistics are collected. Whenever the results of thesmaller group are almost identical to those of the larger group, it is ascertained that a

reasonable standard of accuracy is attained. The term reasonable standard is relative,

depending upon the object of the enquiry and the resources available.

7. Statistics should be collected in a systematic manner

It is essential that statistics must be collected in a systematic manner so that they may

conform to reasonable standards of accuracy.

8. Statistics should be placed in relation to each other and for the purpose of comparison

The data that have been collected for analysis should reflect homogeneous character and be capable of being compared with each other. When the data is of heterogeneous type, it

is not possible to compare the values, thus cannot be placed in relationship to the other.

For example, the height of a person and the success in his business can not be placedtogether because it does not make any sense and thus can not be compared to each other.

2.2 LIMITATIONS OF STATISTICS

Application of Statistics has several limitations. A description of a few limitations is given

below:

1. Statistics does not study qualitative phenomenon

Statistics can be applied only to those problems which are capable of quantitative

expressions. The situations involving characteristics which cannot be expressed in figures

have very little use of statistical methods. For example, the qualitative characteristics suchas Good, Bad, Beauty, Honesty, Pleasure, Joy, Satisfaction etc., are not measurable and

hence can not be expressed in figures. In such cases, statistical methods cannot be of much

help. Therefore, whenever it is possible to relate such qualitative information with otherfactors which are measurable in nature, they may be indirectly expressed as numeric

quantities. For instance, pleasure itself may not be capable of quantitative analysis but

many factors which are related to this phenomenon are capable of being expressed infigures and as such can throw some light on the study of this problem. A study of the

number of tax evaders can indirectly tell us something of the problem under study. Again,

the service rendered by a business firm to its customers can be measured in terms of thekind of service and the number of customers who get utmost satisfaction and if the

number of customers who have received proper service is decreasing, it would be possible

to modify the procedure of rendering service.


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2. Statistics does not reveal all the facts

Statistics cannot reveal all the facts about the population. It is known that many problemsare affected by some factors which are not capable of statistical analysis. Hence, it would

not be possible always to examine a problem in all its dimensions by a statistical approachalone. For instance, in a study relating to the culture or religion of a country, many

problems have to be examined and addressed based on the relevant information about the

background of the country. All these things do not come under the orbit of statistics.

3. Statistical laws are true only on average

Statistics as a science is not accurate as many other sciences are, and statistical methods

are not very precise and correct. Laws of statistics are not true universally and are trueonly on an average. Statistics deal with certain phenomena which are affected by a

multiplicity of causes and it is not possible to study the effects of each of these factorsseparately as is done under experimental methods. Due to this limitation in the statisticalmethods, the conclusions arrived at are not perfectly accurate and consequently the same

conclusions cannot be arrived at under similar conditions at all times.

4. Statistics does not study individuals

For purposes of analysis of statistical data, the aggregates arrived are most often reduced

to single figures. However, statistics deal with aggregates. For instance, an individual item

of a time series data is specifically unimportant; but the series is usually condensed into anaverage for purposes of comparison. Moreover, individual values observed separately do

not constitute statistical data. This is a limitation. It is important to have the group of

individual values, which together have to be analysed to draw conclusions. For instance, it

is important to have the marks scored by all the students in a class in an examination, based on which the decisions are to be made rather than to have the marks of an

individual.

In a similar way, the average income of a group of persons might have remained the sameover two periods and yet many persons in the group might have become poorer than what

they were before. Statistical methods ignore such individual cases. Thus, statistical

methods have no place for an individual item of a series.

5. It is liable to be misused:

Statistics are liable to be misused easily. Statistics is a delicate science and consequently

should be used with caution. There is very great possibility of the misuse of this science as

any type of meaningless conclusion can be drawn from the results arrived from the data.In practice, statistical methods can be properly used only by trained or experienced people.

Lack of experience or training in handling data leads one to make inaccurate results.

Misuses, unfortunately, are probably as common as valid uses of statistics. Hence, it is

more important to discriminate between a valid and an invalid use of statistics and thenknow how to make effective use of statistics.

6. Statistics often leads to false conclusions

It happens, generally, in cases where statistics are quoted without context or details.

Suppose that in a certain competitive examination, the students belonging to one centrehave done better than those of another centre. It does not mean that the first centre has a


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better standard than the other. This is so because there is a possibility that the candidates

in the first centre may have been coached effectively while those of the other centre maynot have trained in that way. Similarly, average expenditure in one hostel may be very

much more than in the other, and on enquiry it may be found that students are generally

spending similar amounts, but in the former hostel the average has been pushed up by astudent or two who may be very rich and spending much more than others.

7. The statistical data must be uniform and its main characteristics must be stable

throughout the study. For example, the wages of labourers in two factories are notcomparable, if the average wage in the first factory is based on wages of adult males and

the average wage in the second factory is based on adult males and adult females. Hence,

it is required that the data must be highly uniform and homogeneous.

8. It is always important to see that statistics must always be handled by experts. Othersare likely to apply wrong methods in statistical analysis.

2.3 SUMMARY

Any concept or theory should possess certain salient features. Statistics, of course, is noexception. In this lesson, the chief characteristics of statistics are described in detail. The

limitations of statistics such as possibility of misuse, of making wrong decisions etc., arealso presented.


1. Consider the score obtained by a student who had taken up a short course in a city

college. What can you say about this course? With this score, can you make anyconclusion?

2. A figure related to sales realized by a firm in a particular month is available. What

kind of conclusion would you draw from this figure?


1. What are the chief characteristics of Statistics?

2. Discuss in detail the serious limitations of statistics with illustrations.


1. Pal, N., and S. Sarkar (2005), Statistics – Concepts and Applications, Prentice –Hall, Englewood Cliffs, NJ, US.


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LESSON-3

PRESENTATION OF DATA

(DIAGRAMS AND GRAPHS)

CONTENTS


3.1. Statistical Diagrams

3.2. Types of Charts and Graphs

3.3. Summary





The aim of this lesson is to emphasize the ways and means of presenting statistical

information through diagrams and graphs. The methods described will help the learner toconstruct the statistical diagrams with ease.

3.1 STATISTICAL DIAGRAMS

The numerical data which are collected for analysis are represented in the form of

diagrams, called statistical diagrams or charts.

Statistical diagrams are generally drawn in order to present data in an attractive andcolourful way and to enable a general perspective of the data to be shown without

excessive detail. Diagrams can be used as a replacement for tabulation of data and often

used for layman to understand somehow the statistical data. They make comparison ofdata much easier and help in establishing trends of the past performance. A complex datacould be made simple and more easily understandable by representing the statistical data

in the form of diagrams.

Besides some advantages as given above, diagrammatic representation of data do have

certain limitations, a few of them are listed below:

• Diagrams may not reveal many facts of data.

• They provide an approximate idea about the characteristics of data.

• Diagrams may not exhibit the minor differences.


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• Sometimes it is more difficult to draw the facts contained in the data from three or

multidimensional diagrams.• Great care must be given in representing data by means of diagrams as they may

often give misleading impressions.

3.2 TYPES OF DIAGRAMS OR CHARTS AND GRAPHS

There are various types of diagrams to represent statistical data. The diagrams can be

classified under the following three categories:

(a) Diagrams to display non-numeric frequency distributions. [Note: Non-numeric frequency distributions describe qualitative characteristics of the data]

(b) Diagrams to display time series.

(c) Miscellaneous diagrams

The first category consists of three types of diagrams, namely, (i) Pictograms, (ii) Simple bar charts and (iii) Pie charts.

In the second category, there are two types of diagrams, namely, (i) Line diagrams and (ii)Simple bar charts.

The diagrams which come under the third category are: (i) Component, percentage andmultiple bar charts and (ii) Multiple pie charts.

Generally diagrams are of one-dimensional, two-dimensional or three dimensional. One-dimensional diagram is a diagram which is constructed on the basis of only one

dimension, namely length. Such type of diagrams is in the form of bars. Simple,component, percentage and multiple bar charts are examples for one-dimensional

diagrams.

Two-dimensional diagram is a diagram which is constructed on the basis of two

dimensions, namely, length and width. Rectangles, squares, circles and Pie diagrams are a

few examples for two-dimensional diagrams.

A detailed discussion of each of the diagrams listed in the three categories (a), (b) and (c)is now presented.

Pictograms

A pictogram is a chart which represents the magnitude of numeric values by using only

simple descriptive pictures or icons. A picture or a symbol or an icon is selected that

easily identifies the data pictorially. It is then duplicated in proportion to the classfrequency, for each class represented. Pictograms are normally used for displaying a small

number of classes, generally with non-numeric frequency distributions. However, they can

be used for representing time series.

The advantage of a pictogram is that it is easy to understand even for laymen; however,there are certain disadvantages, such as, (i) not accurate enough for statistical presentation


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and (ii) symbol magnification, sometimes, may be confusing when the data are not clearly

shown.

Simple Bar Charts

A simple bar chart is a chart consisting of a set of non-joining bars and represents themagnitude of a variable. A separate bar for each time point or class is erected to a height

proportional to the data value or class frequency. The widths of the bars drawn for eachtime or class are always the same. For an attractive and elegant display, each bar may be

shaded or coloured differently.

Simple bar charts can be used to represent non-numeric frequency distributions and time

series equally well.

Simple bar charts are easy to construct and to understand the values being represented by bars. Besides these advantages, simple bar charts have the following special features:

(i) The charts can be drawn with vertical or horizontal bars, but must show a scaled

frequency axis.

(ii) The charts are easily adapted to take into account of both positive and negative

values.

(iii) Two bar charts can be placed back-to-back for comparison purposes.

A procedure for the construction of simple bar chart is given below:

1. Decide whether bars should be vertical or horizontal.2. In the case of vertical bars, take the data values on y – axis and the time point on the

x – axis.

3. Erect the bars to the heights proportional to the data value.

In order to demonstrate this procedure the following illustration is presented:

Example 3.1

Draw a simple bar diagram for the following data relating to profit achieved by a business

firm during 2000 - 2007.

Year

Profit

(in Rs. Lakhs)

2000 10.5

2001 12.3

2002 15.6

2003 19.22004 20.1

2005 19.1

2006 17.72007 16.9


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Solution

The time points (years) are taken along the x – axis and the data values (profit) are taken

along the y – axis. Simple bars are drawn against the years with their heights proportional

to respective profits. Figure 3.1 displays the simple bar diagram constructed in the mannerdescribed.

Figure 3.1

Multiple Bars Charts

Another one-dimensional diagram which represents two or more series of data is referred

to as multiple bars chart. In this chart two or more bars are drawn and each bar is adjoinedwith the other bars representing the values of two or more series. The heights of the bars

are in proportion to the data values in the respective series.

A simple procedure for constructing a multiple bars chart is described below:

1. Take the data values along the y – axis and the time points on the x – axis.

2. Erect the bars to the heights proportional to the data value in each of the given seriesand adjoin them so that that there is no gap between the bars corresponding to eachtime point.

Note

This chart enables one to make comparisons of the data values of different variables in a

series over a given period of time points. Further, it helps to compare the values of the

same variable between two or more series over a period of time.

The following example demonstrates the construction of a multiple bars chart.

Simple Bar Diagram

0

5

10

15

20

25

1 2 3 4 5 6 7 8

Year

Profit


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Example 3.2

The following table presents the details of sales and profit achieved by a business firmduring 2000 - 2007. Draw a simple bar diagram to represent both series of data.

Year

Sales

(in Rs Lakhs)

Profit

(in Rs Lakhs)

2000 125.3 10.5

2001 130.9 12.32002 140.3 15.6

2003 162.8 19.2

2004 168.2 20.12005 161.7 19.1

2006 158.3 17.72007 155.1 16.9

Solution

Here, simple bars are to be drawn representing the two series of data, namely, sales and

profits. It is obvious that corresponding to each time point, two bars need to be

constructed and adjoined. Figure 3.2 displays the simple bar diagram drawn in this way.

Simple Bar Diagrams

0

20

40

60

80

100

120

140

160

180

2000 2001 2002 2003 2004 2005 2006 2007

Year

S a l e s a n d P r o f i t s

( i n R s .

L a k h s )

Sales

Profit

Figure 3.2

Pie Charts

A pie chart, also called as circular diagram, represents the total of a set of components of a

variable using a single circle, called pie. Here, the circle is split into a number of parts

equal to the number of components (i.e., pieces of pie), with the size of each partrepresenting the magnitude of the component, i.e., the size being drawn in proportion to

magnitude of the component. The parts of the circle are separated by straight lines drawn

from the centre to the circumference of the circle. In order to construct a pie chart, the size


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of each part in degrees needs to be calculated. For an elegant display of parts, they can be

shaded or coloured differently.The procedure for constructing a pie chart consists of the following steps:

(i) Calculate the proportion of the total that each component represents by using theformula given below:

componentstheallof valueTotal

component kththeof ValuePk = .

(ii) Multiply each proportion by 360o, giving the sizes of the relevant components (in

degrees) which need to be drawn. That is, obtain the degree to each component by

using the following formula:

Degree = Pk × 360o.

(iii) Compute cumulative degrees.

(iii) Draw a circle with a convenient radius and split the circle into as many parts as equal

to the number of component based on the cumulative degrees.

A pie chart has the merits that it is a more appealing way of presenting data and that the

comparison of classes in relative terms is made easy.

The major demerits of the chart are: (i) the sectors in a circle must be defined carefullyand (ii) compilation of data to each sector is more complex.

Example 3.3

Annual budget allocation for a business firm under various heads of expenditure for the

financial year 2008-09 is given below:

Heads of ExpenditureBudget Allocation

(in Rs. Lakhs)

Salary 100

Purchase 30

Board Meetings 5Travel 7

Reports 2Overhead 5Miscellaneous 10

Total 159

Draw a pie chart.

Solution

A pie chart or circular diagram is constructed by expressing the values of the sectors or

components in terms of degrees taking the whole as 360 degrees. The following table

which presents the component values in terms of degrees and percentages is constructed based on the procedure described earlier:


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Category Rs. inLakhs

Degree Percentage

Salary 100 =× 360159

100226

o100

360

226× =64

Purchase 30 =× 360159

30 68

o100

360

68× =19

Meetings 5 =× 360159

5 11

o100

360

11× = 3

Travel 7 =× 360159

7 16

o100

360

16× = 4

Reports 2 =× 360159

2 5o 100360

5 × = 1

Overhead 5 =× 360159

5 11

o100

360

11× = 3

Miscellaneous 10 =× 360159

10 23

o100

360

23× = 6

Total 159 360o

100

Figure 3.3 is the pie chart which portrays various components in proportion to the degrees

tabulated above.

Pie Diagram

64%

19%

3%

4%

1%

3%

6%

Salary

Purchase

Meetings

Travel

Reports

Ovehead

Miscellaneous

Figure 3.3

Line Diagrams

A line diagram, also known as historigram, plots the values of a time series as a sequence

of points joined by straight lines. The time points are always represented along the

horizontal axis and the values of the variable along the vertical axis.


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The major advantages of line diagrams are as follows:

(i) They are easy to construct and understand.

(ii) They show sense of continuity which is not present in a bar chart.

(iii) They enable direct comparison.

The following are the disadvantages of line diagrams:

(i) The line diagrams might be confusing when many diagrams with closely associatedvalues are compared together.

(ii) No provision to display total figures where several diagrams are displayed.

Example 3.4

For the time series data given in Example 3.1, draw a line diagram.

Solution

Figure 3.4 presents the line diagram drawn from the data on profit for various years bytaking the data values on y – axis and the time points on x – axis.

Line Diagram

0

5

10

15

20

25

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Year

P r o f i t ( i n

R s .

L a k h s )

Figure 3.4

Component, Percentage and Multiple Bar Charts

These charts are used as extensions of simple bar charts, where another dimension of the

data is given. For example, where a simple bar chart might show the production of a

company by year, one of these charts would be used if each year’s production was splitinto, say, export and home consumption, i.e., a component time series.


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In component bar charts, each bar represents a class and splits up into different component

parts. Comparison among different components and comparison between the total and acomponent are made simple by these charts. Components bar charts are also termed as

sub-divided bar charts.

In percentage bar charts, each bar represents a class and all bars are drawn to the sameheight, representing 100% (of the total). The component parts of each class are then

calculated as percentages of the total and shown within the bar accordingly. One may

observe that there is a difference between a component bar chart and a percentage barchart. In a component bar chart, the bars are of different heights as the totals usually

different, whereas in a percentage bar chart, all the bars are of same height as the value of

individual bar is expressed in terms of percentage.

Multiple bar charts have a set of bars for each class with each bar representing a singlecomponent part of the total. Within each set, the bars are physically joined and always

arranged in the same sequence, and sets of bars should be separated.

For all three charts, the components are normally shaded and a legend (key) would be

shown at the side of the chart.

Example 3.5

For the time series data given in Example 3.2, construct a component bar chart.

Solution

A component bar chart is constructed based on the following procedure:

1. Compute the cumulative value of the components of a variable for the given time

points.

2. Corresponding to each time point, draw a simple bar with its height proportional tothe cumulative value of the variable.

3. Sub – divide the bars according to the values of the components.

Using this procedure, the following table is constructed:

Year Sales(in Rs Lakhs)

Profit(in Rs Lakhs)

CumulativeValues

2000 125.3 10.5 135.8

2001 130.9 12.3 143.2

2002 140.3 15.6 155.9

2003 162.8 19.2 182.02004 168.2 20.1 188.3

2005 161.7 19.1 180.8

2006 158.3 17.7 176.0

2007 155.1 16.9 172.0


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Simple bars are drawn with their heights in proportion to the cumulative values presented

in the last column of the above table plotted against the time points in a graph by takingtime points on the x – axis and cumulative values on the y - axis. According to the values

of the two components (sales and profits), each bar is sub-divided into two. Figure 3.5

displays the component bar chart prepared in this manner.

Component Bar Chart

0

50

100

150

200

2000 2001 2002 2003 2004 2005 2006 2007

Year

S a l e s a n d P

r o f i t s

( i n R s . L a k

h s )

Profit

Sales

Figure 3.5

Example 3.6

Various details of two commodities A and B are given below:

Category Commodity A Commodity B

Price per unit Rs. 10 Rs. 15

Number of units sold 100 100

Production Cost Rs. 300 Rs. 500

Cost of components Rs. 500 Rs. 800

Profit Rs. 200 Rs. 200

Construct a component bar chart based on the given data.

Solution

Here, the selling cost of commodity A and commodity B are found as Rs. 1000 and Rs.

1500 respectively. While constructing the component bar chart, it should be ensured that

the bar for each commodity is to account for the corresponding selling cost, which is based on the production cost, components cost and the profit. Thus, for the given data, the

component bar chart is constructed (Figure 3.6), where series 1 represents the cost of the

components, series 2 represents production cost and series 3 represents profit, as shown

below:


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Multiple Bar Diagram

Series1

Series1

Series2

Series2

Series3

Series3

0

200

400

600

800

1000

1200

1400

1600

Commodity A Commodity B

C o s t s ( i n R u p e e s )

Series3

Series2

Series1

Figure 3.6

Example 3.7

The data relating to expenditure in the production of a certain electronic component during

different periods of time are given below:

Category 2005 2006 2007Cost of raw material 10000 11000 13000

Wages 4000 4500 5500

Expenses 1000 1100 1400Overhead expenses 2000 2000 2400

Miscellaneous 1000 1000 1200

Construct a sub-divided bar chart for the given data. Also, compute percentage of all

expenses in each of the year and draw a percentage bar diagram.

Solution

Here, first the total cost of the component should be arrived for each year. While

constructing the sub-divided bar diagram, the vertical bar is erected for each of the given

years and it should account for the associated total cost. Here, the cost of raw material,wages, expenses, overhead expenses and miscellaneous are assumed as series 1, series 2,

series 3, series 4 and series 5 respectively. Thus, for the given data, the sub-divided bar

chart is constructed and displayed as Figure 3.7.


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Sub-divided Bar Diagram

0

5000

10000

15000

20000

25000

2005 2006 2007

Year

E x p e n d i t u r e s Miscellaneous

Overhead expenses

Expenses

Wages

Cost of raw material

Figure 3.7

The percentage bar diagram is formed by expressing the expenses of various categories in

terms of percentages and by drawing bars corresponding to each year. For the given data,

the percentages for each category for the three given years are computed and tabulated

below:

2005 2006 2007Category

Value % Value % Value %

Cost of raw material 10000 55.6 11000 56.1 13000 55.3

Wages 4000 22.2 4500 23.0 5500 23.4Expenses 1000 5.6 1100 5.6 1400 6.0

Overhead expenses 2000 11.1 2000 10.2 2400 10.2

Miscellaneous 1000 5.6 1000 5.1 1200 5.1

Based on the percentages tabulated for each category for the given period of time the

percentage bar diagram is constructed in Figure 3.8.


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Percentage Bar Diagram

0%

20%

40%

60%

80%

100%

2005 2006 2007

Year

P e r c e n t a g e s

Miscellaneous

Overhead expenses

Expenses

Wages

Cost of raw material

Figure 3.8

Multiple Pie Charts

Multiple pie charts can be used as an alternative to percentage bar charts; that is, a piechart (360 degrees) replaces a bar (100%) for each class or year.

The advantage of using multiple pie charts as opposed to percentage bar chart is mainly

visual impact; they are generally felt to be more attractive. However, their construction is

more involved and this is considered as a major disadvantage.

3.3 SUMMARY

Statistical data, in general, are represented by means of diagrams, charts, graphs and

tables. In this lesson, the methods of constructing statistical diagrams such as simple bardiagram, multiple bar diagram, component bar diagram and pie charts are presented.

Illustrations are also given appropriately.


1. Draw a line diagram for the data related to numbers of units of a particular productsold in a store during the first six months in a year.

Month January February March April May June

Sales 12 18 28 23 30 26


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2. The data given below show the amount of cereals (in tons) produced in a particular

region during 2003 – 2007. Draw component bar chart to represent the data.

Amount of Cereals (in tons) duringCereals

2003 2004 2005 2006 2007

Wheat 336 482 500 347 450

Barley 866 856 901 727 866

Oats 131 136 108 122 97

Others 25 23 22 23 18

3. The costs associated with two business projects are given below:

Costs (in Rs. Lakhs.)ProjectSet-up Running Overhead Labour

A 265 420 82 150

B 210 289 65 115

Display the data using a component bar chart. Also draw a pie chart for each of the projects.

4. The following data represents the number of employees in each of five categories ofemployees in a business enterprise. Display the given data by (a) a pie chart and (b)

a simple bar chart.

Category A Category B Category C Category D Category E

Number ofEmployees

35 48 17 22 8

5. The data given below show the production (in tones) of two varieties of a particular

crop during 2000 – 2005. Display the information in a bar chart.

Production of Crops (in tons)Year

Variety A Variety B

2000 42 30

2001 48 352002 29 38

2003 25 31

2004 30 34

2005 34 30


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6. Investments made by a business executive of a company during 2005 – 2007 are

given below:

YearTypes of Investments

2005 2006 2007

Bank Deposits Rs. 30,000 Rs. 45,000 Rs. 58,000

Provident Fund Rs. 50,000 Rs. 54,000 Rs. 60,000

Insurance Premiums Rs. 20,000 Rs. 25,000 Rs. 28,000

Gold Rs. 60,000 Rs. 80,000 Rs. 90,000

Display the information given above using (a) a percentage components chart and (b) a

multiple bar chart.


1. What is a statistical diagram? What purpose a statistical diagram serve?

2. List out various types of charts.

3. What is pictogram?

4. What is line diagram? How do you construct a line diagram?

5. Write down the procedure of constructing a pie chart.

6. What is component bar diagram? How do you construct such a diagram?


1. Pal, N., and S. Sarkar (2005), Statistics – Concepts and Applications, Prentice –

Hall, Englewood Cliffs, NJ, US.

2. Levin, R.I., and D.S. Rubin (1997), Statistics for Management, 7/e, Prentice – Hall,Englewood Cliffs, NJ, US.


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LESSON-4

FREQUENCY DISTRIBUTIONS AND CHARTS

CONTENTS


4.1. Raw Data

4.2. Data Arrays

4.3. A Simple Frequency Distribution

4.4. A Grouped Frequency Distribution

4.5. Pictorial Representation of a Frequency Distribution

4.6. Cumulative Frequency Distributions

4.7. Relative-Frequency Frequency Distributions

4.8. Relative-Cumulative Frequency Distributions

4.9. Summary





This lesson presents the meaning and construction of frequency distribution. The rules for

forming the distribution of data and the corresponding graphical charts are discussed. Thelucid way of presentation of the contents in this lesson will enable one to draw the

frequency polygons, frequency curves, cumulative frequency curves etc., with much ease.

4.1 RAW DATA

Data or information that has not been arranged in any way is called raw data.

Examples

1. The set of ages of 1000 workers in a large industry constitutes raw data.

2. The set of scores of candidates in an entrance examination for admission into a

business school forms raw data.


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Specifically, the raw data related to the number of students who have got admission into

an International Business School from each of the 50 colleges in a city are displayed below:

1 3 2 1 0 2 5 1 2 3

4 0 5 6 1 2 1 2 6 20 1 6 1 6 2 0 4 5 1

5 3 4 1 4 6 7 2 3 5

1 2 4 2 1 3 5 1 6 2

4.2 DATA ARRAYS

An arrangement of raw data in an order of magnitude or in a sequence is called data array.An array, usually called as data array, enables one to extract some information from thedata.

The raw data given above are arrayed and shown below:

0 0 0 0 1 1 1 1 1 1

1 1 1 1 1 1 2 2 2 22 2 2 2 2 2 2 3 3 3

3 3 4 4 4 4 4 5 5 5

5 5 5 6 6 6 6 6 6 7

This array enables one to identify certain information contained within the data set. The

lowest and the highest number are respectively identified as 0 and 7. The number 0 occurs

4 times and the number 7 occurs only once. From these, it is inferred that from 4 collegesno student has got admission and from only one college, a maximum number of students

has been selected.

4.3 SIMPLE FREQUENCY DISTRIBUTION

Raw data sets some times may contain a limited number of values, with each value may

occur many numbers of times. In such a case, the raw data may be organized in a formtermed as a simple frequency distribution. A simple frequency distribution, also called as

frequency table, is a tabular arrangement of data values together with the number ofoccurrences, called frequency, of such values. The structure of a frequency table is

normally applicable to discrete raw data, since data values are quite likely to be repeated

many times and is not normally suitable for continuous data.

Formation of a Simple Frequency Distribution

A simple frequency distribution is formed using a tool called as ‘tally chart’. A tally chart

is constructed using the following method:


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(a) Examine each data value.

(b) Record the occurrence of the value with the symbol (|), called as tally mark.

(c) Find the frequency of the data value as the total of tally marks corresponding to that

value.

(d) Arrange the data values along with frequencies in a tabular form. Such a tabular

arrangement is said to be a simple frequency distribution.

Example 4.1

Consider the data related to number of students admitted into a Business School given inearlier example. It is identified that the lowest number is 0 and the highest number is 7. As

the data values are discrete in nature, a simple frequency distribution using tally marks is

obtained as follows:

Data Value Tally Marks Total

0 |||| 4

1 ||||| ||||| || 122 ||||| ||||| | 11

3 ||||| 5

4 ||||| 5

5 ||||| | 66 ||||| | 6

7 | 1

Total 50

4.4 GROUPED FREQUENCY DISTRIBUTION

It is necessary to summarize and present large mass of data in useful ways so that

important facts from the data could be extracted and effective decisions could be drawn. A

large mass of data is summarized in such a way that the data values are distributed intogroups, or classes, or categories. This enables one to determine class frequencies, defined

as the number of values lying in each class. .

A standard form into which the large mass of data is organised into classes or groupsalong with the frequencies is known as a grouped frequency distribution. A grouped

frequency distribution is defined as a tabular arrangement of data values by various classes

or groups together with the corresponding class or group frequencies.

Example 4.2

The following table displays the number of orders received by a business firm each week

over a period of one year.

The table is a grouped frequency distribution in which the numbers of orders are given as

class intervals and number of weeks as frequencies.


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Number of ordersreceived

Numberof weeks

0 – 4 2

5 – 9 8

10 – 14 1115 – 19 14

20 – 24 6

25 – 29 430 – 34 3

35 – 39 2

40 – 44 1

45 – 49 1

Terms under Frequency Distributions

In a grouped frequency distribution, the class or group of data values is said to be the classinterval. For example, the ages of workers may be given in a group such as 20 – 30. Here,20 – 30 is said to be the class interval. The lower and upper values of each class interval

are called the class limits.

The lower and upper values of a class that has common points between classes are calledclass boundaries. The class boundaries are specified in such a way that the upper boundary

of one class coincides with the lower boundary of the next class. In a frequency

distribution, when there is a difference between the upper value of one class and the lowervalue of the next class, the class boundaries are fixed by adding 0.5 with the upper limits

and subtracting 0.5 with the lower limit. Alternatively, the class boundaries are found by

adding the upper limit of one class to the lower limit of the next class and dividing it by 2.

The width or length of a class is defined as the numerical difference between lower andupper class boundaries (and not class limits). It is also called as the size of the class.

Class mid-points are situated in the centre of the classes and are called class marks. They

can be identified as being mid way between the upper and lower boundaries (or limits).

A particular use of class mid points is to estimate the totals of all the items lying in theclass. This can be done by multiplying the class mid-points with the class frequency.

Thus, if a class is described as 10 to 20 (mid-point 15) with a frequency of 6, an estimateof the total of all the items in the class is 15 x 6 = 90.

Certain Remarks on Compilation of Grouped Frequency Distributions

(a) The values given in the data set must be contained within one (and only one) class.

Thus overlapping classes must not occur. Also, the combined set of classes must

contain all items. For instance, the set of classes 10-14, 15-19, 20-24 etc., would besuitable for data measured as whole numbers, but would not be suitable for data

measured to one decimal place, since, for example, there is no provision for

accommodating the value 14.6 in the above structure.


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(b) The classes must be arranged in the order of their magnitude.

(c) Normally, in total, 8 to 10 class intervals in a frequency distribution may be defined.It is not desirable to have less than 5 or more than 15 class intervals. It is to be noted

when there are very few classes, one may have a good overall summary of the natureof the data and when there are many classes, more information is generated to

comprehend quickly the overall nature of the data.

(d) Class intervals should be defined in such a way as to assimilate easily with ranges

that naturally describe the data being presented.

(e) Frequency distributions having equal class widths throughout are preferable. When

this is not possible, classes with smaller or larger widths can be used. Open ended

classes are acceptable but only at the two ends of a distribution.

Formation of a Grouped Frequency Distribution

To summarize raw data in a logical way, a frequency distribution is formed. The following

procedure is adopted to form a grouped frequency distribution.

Step 1: Determine the range of values covered by the data as the difference between the

largest and the smallest values. (Any extreme values present at either end of the data aresometimes ignored).

Step 2: Divide the range by the number of class intervals to obtain a standard class width.

(If, for instance, 10 classes are required, the range should be divided by 10).

Step 3: Determine the frequencies of each class interval by using a tally chart.

Step 4: Tabulate the class intervals together with the corresponding frequencies. The

resulting table is called the frequency distribution.

Note

1. It should be noted that in a frequency distribution, the first class should contain thelowest value and the last class should contain the highest value.

2. The number of class intervals may be determined by using the followingmathematical formula, (called Sturges formula):

k = 1 + 3.322log10 N,

where N is the total frequency and k is the number of class intervals.


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Example 4.3

The data related to the number of orders received by a business firm each week over a

period of one year are given below:

20 38 43 16 19 7 10 13 5 29 17 13

2 10 21 37 25 19 23 32 17 17 22 27

10 4 11 16 16 24 22 31 46 18 14 9

15 5 6 8 12 12 8 6 18 31 13 14

16 17 18 28

For the given data, construct a grouped frequency distribution.

Solution

1. The lowest and the largest values are observed as 2 and 46 respectively. Hence, therange is obtained as 46 – 2 = 44.

2. Dividing 44 by 10, the class width is obtained as 4.4, which is adjusted to 5.

3. The frequency distribution is now formed with 10 class intervals each of size 5. The

frequencies are computed using tally marks. Thus, the grouped frequencydistribution for the given data is displayed in Table 4.1.

Table 4.1

Frequency Distribution of Number of Orders

Class Intervals Tally Marks Frequencies

0 – 4 || 2

5 – 9 ||||| ||| 8

10 – 14 ||||| ||||| | 11

15 – 19 ||||| ||||| |||| 1420 – 24 ||||| | 6

25 – 29 |||| 4

30 – 34 ||| 3

35 – 39 || 2

40 – 44 | 1

45 – 49 | 1

Total 52


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4.5 PICTORIAL REPRESENTATION OF A FREQUENCY DISTRIBUTION

A frequency distribution can be represented pictorially using (i) a histogram, (ii) a

frequency polygon and (iii) a frequency curve. The meaning and the method ofconstruction of such charts are described below:

Histograms

A frequency distribution can be represented pictorially by means of a histogram. Ahistogram is a chart consisting of a set of vertical bars having their base on a horizontal

axis, and is constructed using the procedure given below:

1. On a two-dimensional graph, represent frequency on the vertical axis and data valueson the horizontal axis.

2. Draw a vertical bar to represent each class interval, with the centre at the class mark,

the bar width corresponds to the class width and the height corresponds to the classfrequency.

3. Join the bars together.

4. Give the appropriate title.

Histograms are helpful to make comparison of two frequency distributions having the

same class structure, when the bars corresponding to each class of the two distributionsare properly drawn and shaded.

Example 4.4

Draw a histogram for a grouped frequency distribution given in Example 4.3.

Solution

A histogram for the given frequency distribution is constructed (i) by taking the class

frequency on y – axis and the variable value on the x – axis, and (ii) by drawing adjacent

vertical bar (rectangle) for each class interval as displayed in Figure 4.1.


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Histogram

0

2

4

6

8

10

12

14

16

0 – 4 5 – 9 10 – 14 15 – 19 20 – 24 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49

Class Intervals

F r e

q u e n c i e s

Figure 4.1

Note

The above procedure is followed when the frequency distribution has equal class intervals.

In the case of a frequency distribution with unequal class intervals, if histogram is

constructed the area of rectangles may not be proportional to the class frequency. Hence,for drawing a histogram adjusted frequency for each class will be calculated and then the

procedure will be adopted. The formula for adjusted frequency is given below:

ervalclassunequalgiventheof Freqeuncyervalclassunequalgiventheof Width

ervalclasslowest theof Width

Frequency Adjusted

intint

int×=

Frequency Polygons and Curves:

A frequency distribution can be represented pictorially using a frequency polygon. A

frequency polygon is a line graph of the class frequency plotted against the class mark and

it is constructed as given below:

(1) Represent each class by a single point with the height of the point showing the class

frequency; the position of the point must be directly above the corresponding class

mid-point.

(2) Join the points by straight lines.


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(3) Label the two axes (horizontal and vertical) appropriately.

(4) Give the appropriate title.

A frequency curve is an approximating curve which is resulted by smoothing the

frequency polygon. Frequency polygons and curves can always be used in place of

histogram, but are particularly useful when there are many classes in the distribution or if

two or more frequency distributions need to be compared. The procedure of constructing afrequency curve for a given frequency distribution consists in the following simple steps:

1. Construct a histogram and frequency polygon based on the data.

2. Smoothen the frequency polygon by drawing smooth line.

Example 4.5

For the data related to number of orders received per week during a year given in Example4.3, draw the frequency polygon and frequency curve.

Solution

A frequency curve is an approximating curve of a frequency distribution. For the

frequency distribution presented in Table 4.1, the frequency polygon and curve are drawnand is displayed in Figure 4.2 and Figure 4.3.

Histogram and Frequency Polygon

0

2

4

6

8

10

12

14

16

0 – 4 5 – 9 10 – 14 15 – 19 20 – 24 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49

Class Intervals

F r e q u e n c i e s

Figure 4.2


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Frequency Curve

0

2

4

6

8

10

12

14

16

0 10 20 30 40 50

Number of Orders Received

N u m b e r o f W e e k s

Figure 4.3

4.6 CUMULATIVE FREQUENCY DISTRIBUTIONS

Cumulative frequency corresponding to a class interval is defined as the total frequency of

all values less than upper class boundary of that class. A tabular arrangement of all

cumulative frequencies together with the corresponding classes is called a cumulativefrequency distribution or cumulative frequency table.

The main difference between a frequency distribution and cumulative frequency

distribution is that in the former case a particular class interval is described according tohow many items lie within it, where as in the later case the number of items which have

values either above or below a particular level is described.

There are two forms of cumulative frequency distributions, which are defined as follows:

(1) Less than cumulative distribution: This consists of a set of item values listed(normally upper boundaries) with each one showing the number of items in the

distribution having values less than the upper boundaries. In this distribution the

cumulative frequencies will be in the ascending order.

(2) More than cumulative distribution: This consists of a set of item values listed

(normally lower boundaries) with each one showing the number of items in the

distribution having values greater than the lower boundaries. In this distribution thecumulative frequencies will be in the descending order.


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Example 4.6

Compute the cumulative frequencies based on the data given in Example 4.3.

Solution

For the data related to the number of orders received per week by a business firm during a

period of one year given in Example 4.3, the less than and more than cumulative

frequencies are computed and displayed in Table 4.2.

Table 4.2

Less than and More than Cumulative Frequency Distributions

Number oforders received

Numberof weeks

Less thanCumulativeFrequencies

More ThanCumulativeFrequencies

0 – 4 2 2 52

5 – 9 8 10 5010 – 14 11 21 42

15 – 19 14 35 3120 – 24 6 41 17

25 – 29 4 45 11

30 – 34 3 48 735 – 39 2 50 4

40 – 44 1 51 245 – 49 1 52 1

Cumulative Frequency Polygons and Ogives

A graph obtained by plotting the cumulative frequencies against the class boundaries (may

be upper or lower) and joining the points with small straight lines is called a cumulativefrequency polygon.

A cumulative frequency curve or ogive curve is an approximating curve, which is resultedon a two-dimensional graph by smoothing the cumulative frequency polygon. The curve

of a less than cumulative distribution, called less than ogive curve, is an increasing curveand has an upward slope from left to right. The curve of a more than cumulative

distribution, termed as more than ogive curve is a decreasing curve and has a downwardslope from left to right.

The construction and the properties of less than ogive and more than ogive curves are

demonstrated in the following illustration:

Example 4.7

For the data given in Example 4.3, draw the ogive curves.


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Solution

The cumulative frequencies are computed using the frequency distribution given inExample 4.3 and are tabulated against the class intervals in Example 4.6.

These cumulative frequencies are plotted on a two dimensional graph. The class intervalsare taken along the horizontal axis and the cumulative frequencies are fixed on the vertical

axis. The less than and more than ogive curves are depicted in Figure 4.4 and Figure 4.5

respectively.

Less than Ogive curve

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BCom-Business Statistics 239